Two infinite families of terminating binomial sums (Q1705223)

The author studies two classes of binomial sums: \[ \sum_{j=2r}^n\left(\frac{j}{j-r}+m-2\right)\binom{j-r}{r}m^{j-1}=m^n\binom{n-r}{r}, \] and \[ (-1)^n+\sum_{j=0}^{n-1}(-1)^j\frac{n+2r+j}{n+r}\binom{n+r}{r+j}2^{n-j-1}=\frac{r}{n+r}\binom{n+r}{r}2^n. \] Both of these are proved using tilings and dominoes. A number of corollaries are presented, some of them give summation formulas for powers of a fixed positive integer. To mention just one very particular summation from the many that can be deduced from the formulas: \[ 2\cdot2^1+3\cdot2^2+\cdots+(n+1)2^n=n2^{n+1}. \]